Strong Approximation of Copulas

. We introduce strong convergence in regard to approximation of copulas. This new type of convergence is useful in dealing with the -product of Darsow, Nguyen, and Olsen for copulas. We also provide tools for constructing strong approximations of copulas by using partitions of unity. Mathematical Subject Classification: Primary 60B10, Secondary 60A10, 60E05, 28E99. 1 Introduction Let I denote the unit interval [0; 1]. Copulas are cumulative distribution functions on I 2 with uniform marginals, more precisely, a copula is a function C(x; y) on I 2 that satisfies 1. (Boundary Conditions) C(x; 0) = C(0; y) = 0; C(x; 1) = x and C(1; y) = y for all x; y 2 I; and 2. (Monotonicity) if 0 x 1 x 2 1 and 0 y 1 y 2 1, then C(x 2 ; y 2 ) \Gamma C(x 1 ; y 2 ) \Gamma C(x 2 ; y 1 ) +C(x 1 ; y 1 ) 0: The idea of a copula was introduced by A. Sklar in response to a question from M. Fr'echet. Sklar proved (see [12, 13]) that if H was the joint distribution function of two random variables, ..

Copula Theory: An Introduction

Abstract In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applie

Copula theory: an introduction

In this survey we review the most important properties of copulas, several families of copulas that have appeared in the literature, and which have been applied in various fields, and several methods of constructing multivariate copulas